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8   COMPLETE HUBERT ALGEBRAS        361

Since 0?i2)* = e*e* = e2ei> it is clear that (b) and (c) are equivalent.
If (^|^2) = 0 then by (15.7.5.2) and (15J.5.5) we have 0 = (<?2|<?2) =
(e1e2\e1e2) = Ik^ll2* so tnat (a) implies (b). Conversely, if ee2 = 0, then
(*i I e2) = (el I e2> = Oi I ^1^2) = 0, and therefore (b) implies (a).

(15.8.5)    Every left ideal I ^ {0} in A contains a self-adjoint idempotent ^0.

Let x be a nonzero element of I. Then z = x*x ^ 0 (15.7.5.7), and z is a
self-adjoint element of I. Multiplying z by a suitable nonzero real scalar, we
may assume that ||/(z)|| = 1, where U is the regular representation of A
(15.8.1). Hence, by (15.8.1) and (11.5.3), ||tf(z2)||=l, and therefore
|| (7(z2M)|| = 1 by induction on n. On the other hand, we have

\\U(zk+l)\\ = \\U(z)U(zk)\\  \\U(z)\\  ||<7(z*)|| = ||t/(z*)||

for all A:, hence the sequence (||C/(zk)||) is decreasing. Since it has infinitely
many terms equal to 1, it follows that || U(zk)\\ = 1 for all integers k, and hence
that ||z*|| ^ If || U\\ for all k. We shall show that the sequence (z2fc) is a Cauchy
sequence in the Hilbert space A. Let n, p be two integers >0, and put
m = n+p. Then

(z2m | z2n) = (z2pz2n | z2") = (zp+2w | zp+2n) = || U(zp) * z2M||2

and

(z2m | z2m) = || U(zp) - zp+2w||2 ^ (zp+2n | zp+2n) = (z2m | z2")

so that, for all ra > ,

I/ 1| l/||2 g (z2m | z2m) ^ (z2m I z2w) g (z2" I z2n).

This shows that the sequence (||z2n||2) is decreasing and has a limit a> 0;
moreover, we have

||z2m _ Z2n||2 = (z2m | ^ _ 2(z2m | ^ + (z2 | ^

^ ||z2n||2 - a

which proves that (z2k) is a Cauchy sequence, as asserted. Hence the sequence
(z2fe) has a limit e. By continuity, e2 = Iim(z4k) = e, and e* = lim(z*)Zfc = e

k-* oo                                                      fc--oo

because z is self-adjoint, and finally ez2 = Iimz2fe+2 = e, so that eel. Finally,

fc~>00

since ||zfc|| ^ I/ 1| U\\ for all fc ^ 1, we have ||e|| > 0, and the proof is complete.

t

A self-adjoint idempotent e ^ 0 is said to be reducible if there exist
two orthogonal self-adjoint idempotents ei9 e2, each nonzero, and such
that e^e^+62- In this case, by (15.8.4), we have ee1=e1e = el and
ee2 = &2 e = e2  If e ^ 0 is not reducible, it is said to be irreducible.ary elements of A is bounded.